Energy-optimal scheduling with variable processing speed: The role of task size variability

In this paper, we study the execution of a single task with an unknown size on a server with variable processing speed. Our goal is to analyze structural properties of the optimal energy consumption under the optimal speed profile that minimizes the expected energy consumption while meeting a hard deadline constraint. Specifically, we investigate how the task size probability distribution impacts the overall energy.

Under mild assumptions, our main result shows that the expected energy consumption induced by the optimal speed profile preserves the convex increasing order with respect to the task size distribution. Then, we leverage this property to derive simple bounds and conduct a worst-case analysis. In particular, we derive a simple, general formula for the energy gap induced by the ‘best’ and ‘worst’ task size distributions, expressed in terms of the support and expectation of the task size.